Cosine of a matrix

The cosine of a matrix $ A $ can be defined through the Taylor series expansion of cosine, adapted for matrix operators: $$ \cos(A) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} A^{2k} $$

In practice, calculating the cosine of a matrix involves using this infinite series, where each term includes an even power of matrix $ A $ along with corresponding coefficients.

A practical example

Consider the matrix $ A $:

$$ A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $$

This matrix represents a rotation and is structured simply enough to allow straightforward calculations.

To find $ \cos(A) $, we’ll use the initial terms of the Taylor/MacLaurin series.

$$ \cos(A) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} A^{2k} $$

Expanding the series for $ \cos(A) $:

$$ \cos(A) = I - \frac{A^2}{2!} + \frac{A^4}{4!} - \dots $$

To simplify, let’s calculate $ A^2 $ for this matrix:

$$ A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -I $$

where $ I $ is the identity matrix.

Note: The power of a matrix is found by multiplying matrix $ A $ by itself, not by squaring each individual element. To compute $ A^2 $, we perform the matrix multiplication $ A \cdot A $: $$ A^2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $$ $$ A^2 = \begin{pmatrix} 0 \cdot 0 + 1 \cdot (-1) & 0 \cdot 1 + 1 \cdot 0 \\ -1 \cdot 0 + 0 \cdot (-1) & -1 \cdot 1 + 0 \cdot 0 \end{pmatrix} $$ $$ A^2 = \begin{pmatrix} 0 - 1 & 0 + 0 \\ 0 + 0 & -1 + 0 \end{pmatrix} $$ The result is: $$ A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} $$ This is equivalent to $-I$, where $ I $ is the identity matrix. $$ A^2 = -1 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = -1 \cdot I $$ $$ A^2 = -I $$

Since $ A^2 = -I $, we can now compute $ A^4 $:

$$ A^4 = A^2 \cdot A^2 = (-I) \cdot (-I) = I $$

With these results, we can simplify the series for cosine:

$$ \cos(A) = I - \frac{A^2}{2!} + \frac{A^4}{4!} - \dots $$

By substituting $ A^2 = -I $ and $ A^4 = I $:

$$ \cos(A) = I - \frac{-I}{2!} + \frac{I}{4!} - \dots $$

$$ \cos(A) = I \left(1 + \frac{1}{2!} - \frac{1}{4!} + \dots \right) $$

Adding up the first few terms, and observing the repeating pattern, gives a good approximation for $ \cos(A) $:

$$ \cos(A) \approx I \left(1 + \frac{1}{2!} - \frac{1}{4!} + \dots \right) $$

This approach converges quickly, provides a practical representation of $ \cos(A) $.

And so forth.